two dimensional motion (projectile motion)

a projectile is an object that moves under the influence of gravity alone. this results in a curved path called a parabolic trajectory.

projectile motion is analysed by resolving the motion into two independent components at right angles to each other.

horizontal component of motion

• the horizontal component of velocity is written as v_x
• there is no horizontal acceleration if air resistance is neglected
• this means v_x remains constant throughout the motion
• horizontal displacement can be found using x = v_x t

vertical component of motion

• the vertical component of velocity is written as v_y
• the vertical motion is affected by gravity
• the acceleration is the acceleration of free fall, g
• g acts downwards and has a value of approximately 9.81 ms⁻²

the horizontal and vertical motions are independent, but they share the same time of flight.

resolving the initial velocity

• if the projectile is launched with speed u at an angle θ
• horizontal component: u_x = u cos θ
• vertical component: u_y = u sin θ

motion equations used in projectile problems

the standard equations of motion are applied separately to the horizontal and vertical directions.

• vertical velocity: v_y = u_y + at
• vertical displacement: s = u_y t + ½at²
• horizontal displacement: x = v_x t

maximum height

• at maximum height, the vertical velocity is zero: v_y = 0
• this condition is used to calculate the time to reach maximum height
• the vertical displacement equation is then used to find the maximum height

time of flight

• the time to reach maximum height is half the total time of flight
• total time of flight: t_total = 2t_max

horizontal range

• the horizontal range is the total horizontal distance travelled
• it is calculated using x = v_x t_total
• the constant horizontal velocity simplifies the calculation